From [1]}, [2]}, and [3]}, it follows that the vector spaces \(H^{\scriptscriptstyle (}k\) )\(\) X\( can be glued together to form a vector bundle \) H\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)  Tg\( with a projectively flat connection in the associated projective bundle. We denote this projectively flat vector bundle by\begin{equation}(\mathbb {P}\mathcal {H}^{\scriptscriptstyle (}k\scriptscriptstyle )\end{equation}\) , \(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) ) Tg.

The action of \(\Gamma _g\)  on \(\mathcal {T}_g\)  lifts to
an action on the projective bundle \(\mathbb {P}\mathcal {H}^{\scriptscriptstyle (}k\) )\(\)  Tg\( which preserves the connection \)\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)\(.Consequently, upon passing to the quotient by this action we obtain the projectively flat vector bundle\begin{equation}(\mathbb {P}\mathcal {\check{H}}^{\scriptscriptstyle (}k\scriptscriptstyle )\end{equation}\) , \(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) ) Mg.

Once a base point in \(\mathcal {M}_g\)  is chosen, the monodromy of () gives rise to the quantum representation

\(\rho ^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
